Kinetics of multi substrate enzyme catalysed reactions Cleland Nomenclature for Enzymes Cleland has devised a standardized way of referring to bisubstrate (Bi-Bi) enzymatic reactions, which make up 60% of all enzymatic transformations. The substrates, products and stable enzyme forms are denoted as follows: Substrates are lettered A, B, C and D, in the order that they are added to the enzyme Products are lettered P, Q, R and S, in the order that they leave the surface of the enzyme Stable enzyme forms are lettered E, F and G, in the order that they occur The number of reactants in the reaction are designated by the terms Uni, Bi, Ter and Quad These are transfer reactions so can be presented as AX + B BX + A

Sequential bi-bi The first important type of bi-bi reaction is known as sequential, which means that all substrates must add to the enzyme before any reaction takes place The sequential bi-bi can be random, any substrate can bind first to the enzyme and any product can leave first ordered, meaning that the substrates add to and products leave the enzyme in a specific order A ternary complex (E + both substrates) is formed in both cases

Sequential bi-bi E.A AX E.AX.B E.A.BX E.AX A BX E-BX

Ping-pong bi-bi (double-displacement) One substrate bind first to the enzyme followed by product P release Typically, product P is a fragment of the original substrate A The rest of the substrate is covalently attached to the enzyme E, which we now designate as F Now the second reactant, B, binds and reacts with the enzyme to form a covalent adduct with the covalent fragment of A still attached to the enzyme to form product Q This is now released and the enzyme is restored to its initial form, E

Steady state kinetics-1 The general rate equation of Alberty (1953) Many two-substrate reactions obey the MM equation with respect to one substate at constant concentration Vmax : max vo when both AX and B are saturating : [AX] which gives 1/2Vmax when B is saturating : [B] which gives 1/2Vmax when AX is saturating : dissociation constant for E + AX EAX

Steady state kinetics-2 At very large [B]: At constant but non saturating [B]: It works well for reactions using 1 or 2 substrate and producing 1 or 2 products but for more complex reactions, other approaches are used

General rate equation of Dalziel (1957)  terms: kinetic coefficients found from primary and secondary plots Primary plots of [E]/v versus 1/[AX] at constant [B] are drawn for series of different [B] Secondary plots: Slope vs 1/[B]  intercept: AX, slope: AXB Intercepts vs 1/[B]  intercept: 0, slope: B

King and Altman procedures (1956) Extremely useful when the mechanisms involved in calculating kinetic constants are laborious Example: Ordered bi-bi

The use of primary plots Lineweaver-Burk plots can be plotted at varying [A] and different fixed values of [B] Ordered and random-sequential mechanisms can be distinguished from ping-pong mechanism, BUT NOT from each other by the use of primary plots Competitive inhibitors or isotope exchange studies are used to differentiate these two mechanisms

Use of inhibitors If there is an available competitive inhibitor for one of the substrates, addition of this compound will slow the overall forward rate and can allow the determination of the exact mechanism Inhibitor can be a dead-end inhibitor or the product inhibitor For an enzyme that requires 2 substrates, a competitive inhibitor of one of the substrate binding sites will display the behavior of competitive, noncompetitive and even uncompetitive inhibitor, depending on: which substrate is varied if inhibitor is reversible dead-end or product inhibitor the mechanism of substrate interaction with enzyme

Dead-end inhibition patterns for a bi-bi reaction Cleland has formulated a series of rules which enable the inhibition patterns for a particular mechanism to be predicted Mechanism Competitive inh. for substrate Inhibitor pattern observed For varied [A] For varied [B] Compulsory ordered A binding first A C N B U B binding first Random ordered Ping-pong

Isotope exchange studies-1 Rate of exchange between a radiolabeled substrate and a product under equilibrium conditions First simple test: if exchange occurs between a substrate and a product when enzyme (+) but second substrate (-)  ping-pong mechanism... e.g. Sucrose phosphorylase Isotope exchange btw sucrose (S1) and fructose (P1) (no S2 and P2) Sucrose fructose Pi G-1-P E E E.sucrose E.glucose.fructose E-glucose E.glucose-1-P

Isotope exchange studies-2 General group transfer reaction: AX + B A + BX Equilibrium constant: Procedure: Add a small amount of radioactively labelled B Measure rate of formation of BX Increase the concentrations of A and AX, keeping [A]/[AX] constant The equilibrium concentrations of B and BX will remain unchanged but the rate of isotope exchange will be affected

Isotope exchange studies-3 B AX A BX E E EB E.B.AX E.BX.A EBX Slight increase in [A] & [AX] may increase the rate of isotope exchange but substantial increase will force the formation of E.B.AX and E.BX.A, making it more difficult for B to dissociate from EB and BX from EBX  exchange rate will reduce AX B BX A EAX E.AX.B E.A.BX EA Free enzyme forced to EAX and EA forms: EAX reacts with B and EA does not affect the initial velocity of liberation of BX from E.A.BX  exchange rate will increase

Binding of ligands to proteins Binding of more than one ligand to an oligomeric receptor (or enzyme in our case) may occur sequentially with binding constants that may not be equal The fractional saturation of such binding sites is described by the Adair equation (Gilbert Adair, 1924) ES denotes enzyme-substrate complex and numbers (0, 1, 2, etc) are the number of substrate attached Such a description don't say how it happens, i.e. why the first binding constant is weak and the second is strong Thus, the Adair equation provides no physical insight as to why various microscopic dissociation differ from each other

Adair equation

Adair equation No interaction between binding sites: Dimer: M2 + S  M2S (binding constant Kb1) Protomer: M + S  MS (binding constant Kb) Forward: dimer has two binding sites so ligand is 2 times more likely to bind Reverse: in both cases, there is only one site that S can dissociate  for overall reaction: Kb1=2Kb M2S + S  M2S2 (Kb2) Forward: both dimer and protomer have 1 free binding site Reverse: two S can dissociate from dimer, only 1 from protomer  for overall reaction: Kb2=1/2 Kb If we substitute these to the general equation: Identical for a protein with a single binding site...

Cooperativity If there are more than one binding sites, there is a possibility of interaction btw the binding sites  cooperativity Positive cooperativity Negative cooperativity Homotropic cooperativity Heterotropic cooperativity Allosteric inhibition: negative heterotropic Allosteric activation: positive heterotropic

A schematic presentation of positive cooperativity

Models of allosteric behavior Sequential model (Koshland, Némethyl & Filmer (KNF), 1966) Subunit interface is changed. Binding of substrate to one active site causes T to R transition and affinity of other subunit for substrate is increased due to substrate interface being altered

Models of allosteric behavior Velocity equation for KNF model Ks: dissociation constant For positive cooperativity  a < 1.0 The model can be extended for a tetrameric enzyme and in this case, the effect of second and third substrate binding to KS is given by: abKS & abcKS S Ks +S aKs

Models of allosteric behavior Concerted transition or symmetrical model (Monod, Wyman & Changeux (MWC), 1965) In a protein, all of the protomers are in the same conformational state: all must be in R- or T-form, no hybrids... Two conformational forms are in equilibrium in favor of T-form in the absence of the ligand Binding of the ligand shifts the equilibrium

Models of allosteric behavior Velocity equation for MWC model Allosteric constant: Assume an enzyme in which the T state has no affinity at all for the substrate and has h number of ligand binding sites KS

Models of allosteric behavior The currently accepted model for allosteric inhibitors and activators is based on the concerted model (MWC) Inhibitors lock all subunits in the T- form Activators lock all subunits in the R-form The MWC model is useful to understand positive homotrophic cooperativity BUT can not explain the negative homotrophic cooperativity. KNF sequential model is usually used for that

MWC vs KNF

Models of allosteric behavior

A more general, simple equation In the case of high cooperativity.... Y (or θ) = fractional saturation h is the Hill constant and n is total number of substrate binding sites or

Hill Plot When the cooperativity is moderate (that is in reality), the experimental data often still well modelled by this equation BUT h will no longer be equal to the number of binding sites (h may not be an integer) Possibilities: h =1.0 no cooperativity (same as the MM equation) n > h > 1.0 positive cooperativity h =n completely cooperative h < 1.0 negative cooperativity

Hill Plot At θ <0.1 and >0.9  slope approaches to 1 (i.e. no cooperativity) Hill coefficient is calculated from the central linear portion of the graph In case MM assumptions valid: v0 is proportional to [ES] and

Binding of oxygen to Haemoglobin In 1904, Bohr and coworkers Fractional saturation of Hb with O2 vs pO2  sigmoidal curve In 1909, Hill explained this on the basis of interaction of binding sites He assumed complete cooperativity and derived Hill equation He found h=2.8 for HB In 1925, Adair developped more general equation for ligand binding In 1960, X-ray data: binding sites are quite apart  cooperativity should be the result of interaction of subunits

Some Facts Oxygen has low solubility in blood (0.1mM) Whole blood, which contains 150 g Hb/L, can carry up to 10 mM oxygen Invertebrates can have alternative proteins for oxygen binding, including hemocyanin, which contains Cu and hemerythrin, a non-heme protein On binding oxygen, solutions of Hb change color to bright red Solutions of hemocyanin (most molluscs, and some arthropods) and hemerythrin change to blue and pink-violet colored, respectively Some Antarctic fish don't require Hb since oxygen is more soluble at low temperature 

Hemoglobin and Myoglobin Structure Hemoglobin of higher vertebrates is made up of two types of chains, referred to as  and β The hemoglobin molecule is a 2-2 tetramer Their primary structures are compared with that of myoglobin The  and β sequences have considerable similarity to one another and some similarity to that of myoglobin The myoglobin and hemoglobin chains have very similar tertiary structures Protein structure affects ligand binding  CO binds 20,000x better than O2 to heme but only 200x better to Mb

Hemoglobin and Myoglobin Relation to Hill Plot In the deoxy conformation  the binding initially occurs along the line corresponding to the weak-binding state But partial oxygenation favors transition to the strong-binding oxy state As oxygen is bound, more and more of the remaining available sites are in hemoglobin molecules that have this conformation The binding curve passes over to that for the strong-binding state

Effect of O2 Binding Heme is distorted into a dome shape and the axis of His is tilted by about 8° When oxygen binds, it flattens the heme Both the His and Val are too close to the heme... His changes its orientation toward the perpendicular. This movement distorts and weakens the whole complex of H bonds and salt bridges In the simplest terms, the binding of O2 pulls the iron a fraction of a nanometer into the heme, producing a lever effect which results in a much larger shift in the surrounding structure, particularly at the critical interfaces

Hemoglobin T-R Shift

Hemoglobin Oxygen Binding Curve

Allosteric Enzymes If you examine the Michaelis Menten equation you will find that an increase in v from 0.1 to 0.9 Vmax requires an 81-fold change in substrate concentration. In other words the velocity is rather insensitive to substrate concentration In allosteric enzymes, however, a small change in one parameter, e.g. substrate, inhibitor, activator concentration, brings about a large change in velocity A consequence of a cooperative system is that the v vs. S plot is no longer hyperbolic Most allosteric enzymes are oligomeric They are generally located at or near branch points in metabolic pathways, where they are influential in directing substrates along one or another of the available metabolic paths

Allosteric Enzymes 9-X  80-X  3-X  When h = 1, need ~80-fold  in [S] to  V0 by 9-fold When h = 4, only need 3-fold  in [S] When h < 1, don’t reach 90% of Vmax even at 1000-fold 

Some Examples Threonine deaminase in E.coli (Abelson, 1954) Addition of isoleucin inhibited the formation of itself Substrate: thereonine and inhibitor: isoleucine bind to different sites

Some Examples Aspartate transcarbamoylase (ATCase) Aspartate carbamoyl transferase from E.coli is the first enzyme in which the active and regulatory sites were shown to be clearly separated They are even in different subunits... Most allosteric enzymes are oligomers consisting of identical subunits Aspartate + carbamoyl phosphate  N-carbamoyl aspartate + Pi First reaction leading the biosynthesis of pyrimidine nucleotides (UMP, UDP, UTP and CTP) Repressors: Uracil and CTP (metabolic end-product) Activator: ATP

Some Examples Aspartate transcarbamoylase (ATCase) 1962: CTP  and ATP  sigmoidal behavior of the curve 1965: catalytic (c) and regulatory (r) sites are in different subunits 1968: sequence analysis and X-ray  c3(r2)3c3 and structural integrity maintained by Zn2+ ions ATP and CTP are competing for the same binding site 1990: site-directed mutagenesis and X-ray  each catalytic subunit has separate domains for aspartate and carbamoyl phosphate binding In T-form, binding sites are further apart In T-form, two catalytic trimers are close to each other, hindering the access to active sites T to R change is in concerted (symmetrical) fashion

Some Examples Aspartate transcarbamoylase (ATCase) T state Key catalytic residues too far apart Key catalytic residues in proper positions R state

Some Examples Phosphofructokinase When a multi-subunit enzyme is fully in the active form, it approximates Michaelis-Menten kinetics (hyperbolic curve)